Properties

Label 384.10.d.a.193.4
Level $384$
Weight $10$
Character 384.193
Analytic conductor $197.774$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(197.773761087\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.4
Root \(-90.8862i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.10.d.a.193.5

$q$-expansion

\(f(q)\) \(=\) \(q-81.0000i q^{3} +2178.24i q^{5} -7658.16 q^{7} -6561.00 q^{9} +O(q^{10})\) \(q-81.0000i q^{3} +2178.24i q^{5} -7658.16 q^{7} -6561.00 q^{9} -61495.3i q^{11} -28283.5i q^{13} +176438. q^{15} +193168. q^{17} +366846. i q^{19} +620311. i q^{21} -820746. q^{23} -2.79162e6 q^{25} +531441. i q^{27} +4.44100e6i q^{29} +2.59982e6 q^{31} -4.98112e6 q^{33} -1.66813e7i q^{35} -2.14501e7i q^{37} -2.29096e6 q^{39} -3.15678e7 q^{41} -9.42768e6i q^{43} -1.42915e7i q^{45} +3.59591e7 q^{47} +1.82938e7 q^{49} -1.56466e7i q^{51} -3.97195e7i q^{53} +1.33952e8 q^{55} +2.97145e7 q^{57} -9.36289e7i q^{59} +3.86390e7i q^{61} +5.02452e7 q^{63} +6.16083e7 q^{65} -1.80133e8i q^{67} +6.64805e7i q^{69} -1.54189e8 q^{71} -3.79531e8 q^{73} +2.26121e8i q^{75} +4.70940e8i q^{77} +5.21268e8 q^{79} +4.30467e7 q^{81} -5.93034e8i q^{83} +4.20768e8i q^{85} +3.59721e8 q^{87} +1.05828e8 q^{89} +2.16599e8i q^{91} -2.10585e8i q^{93} -7.99080e8 q^{95} +2.91252e8 q^{97} +4.03470e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 13632 q^{7} - 52488 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 13632 q^{7} - 52488 q^{9} - 90720 q^{15} + 8304 q^{17} - 4612608 q^{23} - 4754904 q^{25} - 7499328 q^{31} - 6213024 q^{33} - 23211360 q^{39} - 43518896 q^{41} + 49382016 q^{47} - 74106808 q^{49} + 19030656 q^{55} + 38141280 q^{57} + 89439552 q^{63} - 110270336 q^{65} - 741751296 q^{71} - 1507903440 q^{73} + 1008373440 q^{79} + 344373768 q^{81} + 423468000 q^{87} - 1337034448 q^{89} - 543950208 q^{95} - 904817936 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 81.0000i − 0.577350i
\(4\) 0 0
\(5\) 2178.24i 1.55862i 0.626636 + 0.779312i \(0.284431\pi\)
−0.626636 + 0.779312i \(0.715569\pi\)
\(6\) 0 0
\(7\) −7658.16 −1.20554 −0.602772 0.797913i \(-0.705937\pi\)
−0.602772 + 0.797913i \(0.705937\pi\)
\(8\) 0 0
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) − 61495.3i − 1.26641i −0.773984 0.633205i \(-0.781739\pi\)
0.773984 0.633205i \(-0.218261\pi\)
\(12\) 0 0
\(13\) − 28283.5i − 0.274655i −0.990526 0.137328i \(-0.956149\pi\)
0.990526 0.137328i \(-0.0438513\pi\)
\(14\) 0 0
\(15\) 176438. 0.899872
\(16\) 0 0
\(17\) 193168. 0.560939 0.280470 0.959863i \(-0.409510\pi\)
0.280470 + 0.959863i \(0.409510\pi\)
\(18\) 0 0
\(19\) 366846.i 0.645792i 0.946434 + 0.322896i \(0.104656\pi\)
−0.946434 + 0.322896i \(0.895344\pi\)
\(20\) 0 0
\(21\) 620311.i 0.696021i
\(22\) 0 0
\(23\) −820746. −0.611553 −0.305776 0.952103i \(-0.598916\pi\)
−0.305776 + 0.952103i \(0.598916\pi\)
\(24\) 0 0
\(25\) −2.79162e6 −1.42931
\(26\) 0 0
\(27\) 531441.i 0.192450i
\(28\) 0 0
\(29\) 4.44100e6i 1.16598i 0.812481 + 0.582988i \(0.198116\pi\)
−0.812481 + 0.582988i \(0.801884\pi\)
\(30\) 0 0
\(31\) 2.59982e6 0.505609 0.252805 0.967517i \(-0.418647\pi\)
0.252805 + 0.967517i \(0.418647\pi\)
\(32\) 0 0
\(33\) −4.98112e6 −0.731163
\(34\) 0 0
\(35\) − 1.66813e7i − 1.87899i
\(36\) 0 0
\(37\) − 2.14501e7i − 1.88157i −0.338999 0.940787i \(-0.610088\pi\)
0.338999 0.940787i \(-0.389912\pi\)
\(38\) 0 0
\(39\) −2.29096e6 −0.158572
\(40\) 0 0
\(41\) −3.15678e7 −1.74468 −0.872341 0.488898i \(-0.837399\pi\)
−0.872341 + 0.488898i \(0.837399\pi\)
\(42\) 0 0
\(43\) − 9.42768e6i − 0.420530i −0.977644 0.210265i \(-0.932567\pi\)
0.977644 0.210265i \(-0.0674327\pi\)
\(44\) 0 0
\(45\) − 1.42915e7i − 0.519541i
\(46\) 0 0
\(47\) 3.59591e7 1.07490 0.537450 0.843295i \(-0.319388\pi\)
0.537450 + 0.843295i \(0.319388\pi\)
\(48\) 0 0
\(49\) 1.82938e7 0.453337
\(50\) 0 0
\(51\) − 1.56466e7i − 0.323859i
\(52\) 0 0
\(53\) − 3.97195e7i − 0.691453i −0.938335 0.345726i \(-0.887633\pi\)
0.938335 0.345726i \(-0.112367\pi\)
\(54\) 0 0
\(55\) 1.33952e8 1.97386
\(56\) 0 0
\(57\) 2.97145e7 0.372848
\(58\) 0 0
\(59\) − 9.36289e7i − 1.00595i −0.864301 0.502975i \(-0.832239\pi\)
0.864301 0.502975i \(-0.167761\pi\)
\(60\) 0 0
\(61\) 3.86390e7i 0.357308i 0.983912 + 0.178654i \(0.0571742\pi\)
−0.983912 + 0.178654i \(0.942826\pi\)
\(62\) 0 0
\(63\) 5.02452e7 0.401848
\(64\) 0 0
\(65\) 6.16083e7 0.428084
\(66\) 0 0
\(67\) − 1.80133e8i − 1.09209i −0.837757 0.546043i \(-0.816134\pi\)
0.837757 0.546043i \(-0.183866\pi\)
\(68\) 0 0
\(69\) 6.64805e7i 0.353080i
\(70\) 0 0
\(71\) −1.54189e8 −0.720097 −0.360049 0.932934i \(-0.617240\pi\)
−0.360049 + 0.932934i \(0.617240\pi\)
\(72\) 0 0
\(73\) −3.79531e8 −1.56421 −0.782105 0.623147i \(-0.785854\pi\)
−0.782105 + 0.623147i \(0.785854\pi\)
\(74\) 0 0
\(75\) 2.26121e8i 0.825212i
\(76\) 0 0
\(77\) 4.70940e8i 1.52671i
\(78\) 0 0
\(79\) 5.21268e8 1.50570 0.752851 0.658190i \(-0.228678\pi\)
0.752851 + 0.658190i \(0.228678\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) − 5.93034e8i − 1.37160i −0.727790 0.685801i \(-0.759452\pi\)
0.727790 0.685801i \(-0.240548\pi\)
\(84\) 0 0
\(85\) 4.20768e8i 0.874294i
\(86\) 0 0
\(87\) 3.59721e8 0.673176
\(88\) 0 0
\(89\) 1.05828e8 0.178790 0.0893952 0.995996i \(-0.471507\pi\)
0.0893952 + 0.995996i \(0.471507\pi\)
\(90\) 0 0
\(91\) 2.16599e8i 0.331109i
\(92\) 0 0
\(93\) − 2.10585e8i − 0.291914i
\(94\) 0 0
\(95\) −7.99080e8 −1.00655
\(96\) 0 0
\(97\) 2.91252e8 0.334038 0.167019 0.985954i \(-0.446586\pi\)
0.167019 + 0.985954i \(0.446586\pi\)
\(98\) 0 0
\(99\) 4.03470e8i 0.422137i
\(100\) 0 0
\(101\) 1.38250e9i 1.32197i 0.750401 + 0.660983i \(0.229860\pi\)
−0.750401 + 0.660983i \(0.770140\pi\)
\(102\) 0 0
\(103\) −5.22223e8 −0.457181 −0.228591 0.973523i \(-0.573412\pi\)
−0.228591 + 0.973523i \(0.573412\pi\)
\(104\) 0 0
\(105\) −1.35119e9 −1.08484
\(106\) 0 0
\(107\) 2.56206e9i 1.88957i 0.327695 + 0.944784i \(0.393728\pi\)
−0.327695 + 0.944784i \(0.606272\pi\)
\(108\) 0 0
\(109\) − 4.72077e8i − 0.320327i −0.987090 0.160164i \(-0.948798\pi\)
0.987090 0.160164i \(-0.0512022\pi\)
\(110\) 0 0
\(111\) −1.73746e9 −1.08633
\(112\) 0 0
\(113\) 9.44729e8 0.545073 0.272536 0.962146i \(-0.412138\pi\)
0.272536 + 0.962146i \(0.412138\pi\)
\(114\) 0 0
\(115\) − 1.78779e9i − 0.953180i
\(116\) 0 0
\(117\) 1.85568e8i 0.0915517i
\(118\) 0 0
\(119\) −1.47931e9 −0.676237
\(120\) 0 0
\(121\) −1.42372e9 −0.603796
\(122\) 0 0
\(123\) 2.55699e9i 1.00729i
\(124\) 0 0
\(125\) − 1.82644e9i − 0.669132i
\(126\) 0 0
\(127\) 2.56290e9 0.874209 0.437105 0.899411i \(-0.356004\pi\)
0.437105 + 0.899411i \(0.356004\pi\)
\(128\) 0 0
\(129\) −7.63642e8 −0.242793
\(130\) 0 0
\(131\) 1.68489e9i 0.499863i 0.968264 + 0.249931i \(0.0804081\pi\)
−0.968264 + 0.249931i \(0.919592\pi\)
\(132\) 0 0
\(133\) − 2.80936e9i − 0.778531i
\(134\) 0 0
\(135\) −1.15761e9 −0.299957
\(136\) 0 0
\(137\) 5.71074e7 0.0138500 0.00692501 0.999976i \(-0.497796\pi\)
0.00692501 + 0.999976i \(0.497796\pi\)
\(138\) 0 0
\(139\) 5.87092e9i 1.33395i 0.745081 + 0.666974i \(0.232411\pi\)
−0.745081 + 0.666974i \(0.767589\pi\)
\(140\) 0 0
\(141\) − 2.91269e9i − 0.620594i
\(142\) 0 0
\(143\) −1.73930e9 −0.347826
\(144\) 0 0
\(145\) −9.67357e9 −1.81732
\(146\) 0 0
\(147\) − 1.48180e9i − 0.261734i
\(148\) 0 0
\(149\) 1.06329e10i 1.76731i 0.468136 + 0.883656i \(0.344926\pi\)
−0.468136 + 0.883656i \(0.655074\pi\)
\(150\) 0 0
\(151\) 4.05058e9 0.634046 0.317023 0.948418i \(-0.397317\pi\)
0.317023 + 0.948418i \(0.397317\pi\)
\(152\) 0 0
\(153\) −1.26738e9 −0.186980
\(154\) 0 0
\(155\) 5.66303e9i 0.788054i
\(156\) 0 0
\(157\) 5.91477e8i 0.0776944i 0.999245 + 0.0388472i \(0.0123686\pi\)
−0.999245 + 0.0388472i \(0.987631\pi\)
\(158\) 0 0
\(159\) −3.21728e9 −0.399210
\(160\) 0 0
\(161\) 6.28541e9 0.737254
\(162\) 0 0
\(163\) − 1.71216e10i − 1.89977i −0.312598 0.949886i \(-0.601199\pi\)
0.312598 0.949886i \(-0.398801\pi\)
\(164\) 0 0
\(165\) − 1.08501e10i − 1.13961i
\(166\) 0 0
\(167\) 1.79932e10 1.79013 0.895065 0.445936i \(-0.147129\pi\)
0.895065 + 0.445936i \(0.147129\pi\)
\(168\) 0 0
\(169\) 9.80454e9 0.924565
\(170\) 0 0
\(171\) − 2.40688e9i − 0.215264i
\(172\) 0 0
\(173\) 1.65957e10i 1.40861i 0.709900 + 0.704303i \(0.248740\pi\)
−0.709900 + 0.704303i \(0.751260\pi\)
\(174\) 0 0
\(175\) 2.13787e10 1.72310
\(176\) 0 0
\(177\) −7.58394e9 −0.580785
\(178\) 0 0
\(179\) 1.76363e10i 1.28401i 0.766701 + 0.642004i \(0.221897\pi\)
−0.766701 + 0.642004i \(0.778103\pi\)
\(180\) 0 0
\(181\) 4.78894e8i 0.0331654i 0.999862 + 0.0165827i \(0.00527868\pi\)
−0.999862 + 0.0165827i \(0.994721\pi\)
\(182\) 0 0
\(183\) 3.12976e9 0.206292
\(184\) 0 0
\(185\) 4.67235e10 2.93267
\(186\) 0 0
\(187\) − 1.18789e10i − 0.710380i
\(188\) 0 0
\(189\) − 4.06986e9i − 0.232007i
\(190\) 0 0
\(191\) 2.02911e10 1.10320 0.551601 0.834108i \(-0.314017\pi\)
0.551601 + 0.834108i \(0.314017\pi\)
\(192\) 0 0
\(193\) 6.29052e9 0.326346 0.163173 0.986597i \(-0.447827\pi\)
0.163173 + 0.986597i \(0.447827\pi\)
\(194\) 0 0
\(195\) − 4.99027e9i − 0.247154i
\(196\) 0 0
\(197\) 6.17930e9i 0.292308i 0.989262 + 0.146154i \(0.0466896\pi\)
−0.989262 + 0.146154i \(0.953310\pi\)
\(198\) 0 0
\(199\) −1.98276e10 −0.896255 −0.448128 0.893970i \(-0.647909\pi\)
−0.448128 + 0.893970i \(0.647909\pi\)
\(200\) 0 0
\(201\) −1.45908e10 −0.630516
\(202\) 0 0
\(203\) − 3.40099e10i − 1.40564i
\(204\) 0 0
\(205\) − 6.87623e10i − 2.71930i
\(206\) 0 0
\(207\) 5.38492e9 0.203851
\(208\) 0 0
\(209\) 2.25593e10 0.817838
\(210\) 0 0
\(211\) 5.37775e10i 1.86780i 0.357539 + 0.933898i \(0.383616\pi\)
−0.357539 + 0.933898i \(0.616384\pi\)
\(212\) 0 0
\(213\) 1.24893e10i 0.415748i
\(214\) 0 0
\(215\) 2.05358e10 0.655448
\(216\) 0 0
\(217\) −1.99098e10 −0.609534
\(218\) 0 0
\(219\) 3.07420e10i 0.903097i
\(220\) 0 0
\(221\) − 5.46347e9i − 0.154065i
\(222\) 0 0
\(223\) 4.21444e10 1.14122 0.570609 0.821222i \(-0.306707\pi\)
0.570609 + 0.821222i \(0.306707\pi\)
\(224\) 0 0
\(225\) 1.83158e10 0.476436
\(226\) 0 0
\(227\) 6.21393e10i 1.55328i 0.629944 + 0.776641i \(0.283078\pi\)
−0.629944 + 0.776641i \(0.716922\pi\)
\(228\) 0 0
\(229\) 2.28989e10i 0.550244i 0.961409 + 0.275122i \(0.0887182\pi\)
−0.961409 + 0.275122i \(0.911282\pi\)
\(230\) 0 0
\(231\) 3.81462e10 0.881449
\(232\) 0 0
\(233\) −6.44391e10 −1.43234 −0.716172 0.697923i \(-0.754108\pi\)
−0.716172 + 0.697923i \(0.754108\pi\)
\(234\) 0 0
\(235\) 7.83277e10i 1.67537i
\(236\) 0 0
\(237\) − 4.22227e10i − 0.869318i
\(238\) 0 0
\(239\) −1.85246e10 −0.367246 −0.183623 0.982997i \(-0.558783\pi\)
−0.183623 + 0.982997i \(0.558783\pi\)
\(240\) 0 0
\(241\) −8.59459e10 −1.64115 −0.820575 0.571539i \(-0.806347\pi\)
−0.820575 + 0.571539i \(0.806347\pi\)
\(242\) 0 0
\(243\) − 3.48678e9i − 0.0641500i
\(244\) 0 0
\(245\) 3.98483e10i 0.706581i
\(246\) 0 0
\(247\) 1.03757e10 0.177370
\(248\) 0 0
\(249\) −4.80357e10 −0.791894
\(250\) 0 0
\(251\) − 3.84815e10i − 0.611957i −0.952038 0.305978i \(-0.901016\pi\)
0.952038 0.305978i \(-0.0989835\pi\)
\(252\) 0 0
\(253\) 5.04720e10i 0.774477i
\(254\) 0 0
\(255\) 3.40822e10 0.504774
\(256\) 0 0
\(257\) 1.23429e11 1.76489 0.882447 0.470412i \(-0.155895\pi\)
0.882447 + 0.470412i \(0.155895\pi\)
\(258\) 0 0
\(259\) 1.64268e11i 2.26832i
\(260\) 0 0
\(261\) − 2.91374e10i − 0.388659i
\(262\) 0 0
\(263\) −2.36808e10 −0.305207 −0.152604 0.988287i \(-0.548766\pi\)
−0.152604 + 0.988287i \(0.548766\pi\)
\(264\) 0 0
\(265\) 8.65188e10 1.07771
\(266\) 0 0
\(267\) − 8.57204e9i − 0.103225i
\(268\) 0 0
\(269\) − 5.33379e10i − 0.621085i −0.950560 0.310542i \(-0.899489\pi\)
0.950560 0.310542i \(-0.100511\pi\)
\(270\) 0 0
\(271\) −1.06039e11 −1.19427 −0.597137 0.802139i \(-0.703695\pi\)
−0.597137 + 0.802139i \(0.703695\pi\)
\(272\) 0 0
\(273\) 1.75445e10 0.191166
\(274\) 0 0
\(275\) 1.71671e11i 1.81009i
\(276\) 0 0
\(277\) − 1.39984e10i − 0.142863i −0.997446 0.0714314i \(-0.977243\pi\)
0.997446 0.0714314i \(-0.0227567\pi\)
\(278\) 0 0
\(279\) −1.70574e10 −0.168536
\(280\) 0 0
\(281\) −9.66573e10 −0.924818 −0.462409 0.886667i \(-0.653015\pi\)
−0.462409 + 0.886667i \(0.653015\pi\)
\(282\) 0 0
\(283\) 1.62302e11i 1.50413i 0.659087 + 0.752066i \(0.270943\pi\)
−0.659087 + 0.752066i \(0.729057\pi\)
\(284\) 0 0
\(285\) 6.47255e10i 0.581130i
\(286\) 0 0
\(287\) 2.41751e11 2.10329
\(288\) 0 0
\(289\) −8.12738e10 −0.685347
\(290\) 0 0
\(291\) − 2.35914e10i − 0.192857i
\(292\) 0 0
\(293\) − 1.22365e10i − 0.0969957i −0.998823 0.0484979i \(-0.984557\pi\)
0.998823 0.0484979i \(-0.0154434\pi\)
\(294\) 0 0
\(295\) 2.03947e11 1.56790
\(296\) 0 0
\(297\) 3.26811e10 0.243721
\(298\) 0 0
\(299\) 2.32136e10i 0.167966i
\(300\) 0 0
\(301\) 7.21987e10i 0.506967i
\(302\) 0 0
\(303\) 1.11983e11 0.763237
\(304\) 0 0
\(305\) −8.41653e10 −0.556908
\(306\) 0 0
\(307\) − 1.03830e11i − 0.667116i −0.942730 0.333558i \(-0.891751\pi\)
0.942730 0.333558i \(-0.108249\pi\)
\(308\) 0 0
\(309\) 4.23001e10i 0.263954i
\(310\) 0 0
\(311\) 1.95717e10 0.118633 0.0593167 0.998239i \(-0.481108\pi\)
0.0593167 + 0.998239i \(0.481108\pi\)
\(312\) 0 0
\(313\) 2.36501e11 1.39278 0.696391 0.717662i \(-0.254788\pi\)
0.696391 + 0.717662i \(0.254788\pi\)
\(314\) 0 0
\(315\) 1.09446e11i 0.626330i
\(316\) 0 0
\(317\) 1.54506e11i 0.859369i 0.902979 + 0.429684i \(0.141375\pi\)
−0.902979 + 0.429684i \(0.858625\pi\)
\(318\) 0 0
\(319\) 2.73100e11 1.47660
\(320\) 0 0
\(321\) 2.07527e11 1.09094
\(322\) 0 0
\(323\) 7.08631e10i 0.362250i
\(324\) 0 0
\(325\) 7.89567e10i 0.392567i
\(326\) 0 0
\(327\) −3.82382e10 −0.184941
\(328\) 0 0
\(329\) −2.75380e11 −1.29584
\(330\) 0 0
\(331\) 4.67240e10i 0.213951i 0.994262 + 0.106975i \(0.0341166\pi\)
−0.994262 + 0.106975i \(0.965883\pi\)
\(332\) 0 0
\(333\) 1.40734e11i 0.627191i
\(334\) 0 0
\(335\) 3.92373e11 1.70215
\(336\) 0 0
\(337\) −4.64738e10 −0.196279 −0.0981394 0.995173i \(-0.531289\pi\)
−0.0981394 + 0.995173i \(0.531289\pi\)
\(338\) 0 0
\(339\) − 7.65231e10i − 0.314698i
\(340\) 0 0
\(341\) − 1.59876e11i − 0.640309i
\(342\) 0 0
\(343\) 1.68938e11 0.659027
\(344\) 0 0
\(345\) −1.44811e11 −0.550319
\(346\) 0 0
\(347\) 1.67034e11i 0.618476i 0.950985 + 0.309238i \(0.100074\pi\)
−0.950985 + 0.309238i \(0.899926\pi\)
\(348\) 0 0
\(349\) 4.14205e11i 1.49452i 0.664534 + 0.747258i \(0.268630\pi\)
−0.664534 + 0.747258i \(0.731370\pi\)
\(350\) 0 0
\(351\) 1.50310e10 0.0528574
\(352\) 0 0
\(353\) −3.92533e11 −1.34552 −0.672759 0.739862i \(-0.734891\pi\)
−0.672759 + 0.739862i \(0.734891\pi\)
\(354\) 0 0
\(355\) − 3.35861e11i − 1.12236i
\(356\) 0 0
\(357\) 1.19824e11i 0.390426i
\(358\) 0 0
\(359\) 5.24918e11 1.66789 0.833943 0.551850i \(-0.186078\pi\)
0.833943 + 0.551850i \(0.186078\pi\)
\(360\) 0 0
\(361\) 1.88112e11 0.582953
\(362\) 0 0
\(363\) 1.15321e11i 0.348602i
\(364\) 0 0
\(365\) − 8.26712e11i − 2.43801i
\(366\) 0 0
\(367\) 2.29807e11 0.661250 0.330625 0.943762i \(-0.392740\pi\)
0.330625 + 0.943762i \(0.392740\pi\)
\(368\) 0 0
\(369\) 2.07116e11 0.581561
\(370\) 0 0
\(371\) 3.04178e11i 0.833577i
\(372\) 0 0
\(373\) 5.03584e10i 0.134704i 0.997729 + 0.0673522i \(0.0214551\pi\)
−0.997729 + 0.0673522i \(0.978545\pi\)
\(374\) 0 0
\(375\) −1.47942e11 −0.386323
\(376\) 0 0
\(377\) 1.25607e11 0.320241
\(378\) 0 0
\(379\) 4.42233e11i 1.10097i 0.834846 + 0.550484i \(0.185557\pi\)
−0.834846 + 0.550484i \(0.814443\pi\)
\(380\) 0 0
\(381\) − 2.07595e11i − 0.504725i
\(382\) 0 0
\(383\) 3.96927e11 0.942576 0.471288 0.881979i \(-0.343789\pi\)
0.471288 + 0.881979i \(0.343789\pi\)
\(384\) 0 0
\(385\) −1.02582e12 −2.37957
\(386\) 0 0
\(387\) 6.18550e10i 0.140177i
\(388\) 0 0
\(389\) − 7.68232e11i − 1.70106i −0.525929 0.850529i \(-0.676282\pi\)
0.525929 0.850529i \(-0.323718\pi\)
\(390\) 0 0
\(391\) −1.58542e11 −0.343044
\(392\) 0 0
\(393\) 1.36476e11 0.288596
\(394\) 0 0
\(395\) 1.13545e12i 2.34682i
\(396\) 0 0
\(397\) − 3.41102e11i − 0.689172i −0.938755 0.344586i \(-0.888019\pi\)
0.938755 0.344586i \(-0.111981\pi\)
\(398\) 0 0
\(399\) −2.27559e11 −0.449485
\(400\) 0 0
\(401\) 5.83691e11 1.12728 0.563642 0.826019i \(-0.309400\pi\)
0.563642 + 0.826019i \(0.309400\pi\)
\(402\) 0 0
\(403\) − 7.35318e10i − 0.138868i
\(404\) 0 0
\(405\) 9.37662e10i 0.173180i
\(406\) 0 0
\(407\) −1.31908e12 −2.38284
\(408\) 0 0
\(409\) 7.80713e11 1.37955 0.689773 0.724026i \(-0.257710\pi\)
0.689773 + 0.724026i \(0.257710\pi\)
\(410\) 0 0
\(411\) − 4.62570e9i − 0.00799631i
\(412\) 0 0
\(413\) 7.17025e11i 1.21272i
\(414\) 0 0
\(415\) 1.29177e12 2.13781
\(416\) 0 0
\(417\) 4.75544e11 0.770156
\(418\) 0 0
\(419\) − 5.88488e11i − 0.932769i −0.884582 0.466385i \(-0.845556\pi\)
0.884582 0.466385i \(-0.154444\pi\)
\(420\) 0 0
\(421\) − 2.50950e11i − 0.389330i −0.980870 0.194665i \(-0.937638\pi\)
0.980870 0.194665i \(-0.0623620\pi\)
\(422\) 0 0
\(423\) −2.35928e11 −0.358300
\(424\) 0 0
\(425\) −5.39253e11 −0.801756
\(426\) 0 0
\(427\) − 2.95904e11i − 0.430750i
\(428\) 0 0
\(429\) 1.40883e11i 0.200817i
\(430\) 0 0
\(431\) −1.22079e12 −1.70409 −0.852046 0.523467i \(-0.824638\pi\)
−0.852046 + 0.523467i \(0.824638\pi\)
\(432\) 0 0
\(433\) 2.85938e11 0.390909 0.195455 0.980713i \(-0.437382\pi\)
0.195455 + 0.980713i \(0.437382\pi\)
\(434\) 0 0
\(435\) 7.83559e11i 1.04923i
\(436\) 0 0
\(437\) − 3.01088e11i − 0.394936i
\(438\) 0 0
\(439\) −4.26567e11 −0.548147 −0.274073 0.961709i \(-0.588371\pi\)
−0.274073 + 0.961709i \(0.588371\pi\)
\(440\) 0 0
\(441\) −1.20025e11 −0.151112
\(442\) 0 0
\(443\) − 2.78131e11i − 0.343109i −0.985175 0.171555i \(-0.945121\pi\)
0.985175 0.171555i \(-0.0548790\pi\)
\(444\) 0 0
\(445\) 2.30518e11i 0.278667i
\(446\) 0 0
\(447\) 8.61265e11 1.02036
\(448\) 0 0
\(449\) 6.47195e11 0.751496 0.375748 0.926722i \(-0.377386\pi\)
0.375748 + 0.926722i \(0.377386\pi\)
\(450\) 0 0
\(451\) 1.94127e12i 2.20948i
\(452\) 0 0
\(453\) − 3.28097e11i − 0.366067i
\(454\) 0 0
\(455\) −4.71806e11 −0.516074
\(456\) 0 0
\(457\) 9.87836e11 1.05940 0.529702 0.848184i \(-0.322304\pi\)
0.529702 + 0.848184i \(0.322304\pi\)
\(458\) 0 0
\(459\) 1.02658e11i 0.107953i
\(460\) 0 0
\(461\) 5.23286e11i 0.539616i 0.962914 + 0.269808i \(0.0869603\pi\)
−0.962914 + 0.269808i \(0.913040\pi\)
\(462\) 0 0
\(463\) −9.21613e11 −0.932039 −0.466020 0.884774i \(-0.654312\pi\)
−0.466020 + 0.884774i \(0.654312\pi\)
\(464\) 0 0
\(465\) 4.58705e11 0.454983
\(466\) 0 0
\(467\) − 1.04570e11i − 0.101738i −0.998705 0.0508689i \(-0.983801\pi\)
0.998705 0.0508689i \(-0.0161991\pi\)
\(468\) 0 0
\(469\) 1.37949e12i 1.31656i
\(470\) 0 0
\(471\) 4.79097e10 0.0448569
\(472\) 0 0
\(473\) −5.79758e11 −0.532563
\(474\) 0 0
\(475\) − 1.02409e12i − 0.923036i
\(476\) 0 0
\(477\) 2.60600e11i 0.230484i
\(478\) 0 0
\(479\) 1.93040e12 1.67547 0.837737 0.546074i \(-0.183878\pi\)
0.837737 + 0.546074i \(0.183878\pi\)
\(480\) 0 0
\(481\) −6.06683e11 −0.516784
\(482\) 0 0
\(483\) − 5.09118e11i − 0.425654i
\(484\) 0 0
\(485\) 6.34417e11i 0.520639i
\(486\) 0 0
\(487\) −1.28669e12 −1.03656 −0.518281 0.855210i \(-0.673428\pi\)
−0.518281 + 0.855210i \(0.673428\pi\)
\(488\) 0 0
\(489\) −1.38685e12 −1.09683
\(490\) 0 0
\(491\) − 1.58967e12i − 1.23436i −0.786823 0.617179i \(-0.788276\pi\)
0.786823 0.617179i \(-0.211724\pi\)
\(492\) 0 0
\(493\) 8.57860e11i 0.654042i
\(494\) 0 0
\(495\) −8.78857e11 −0.657953
\(496\) 0 0
\(497\) 1.18080e12 0.868109
\(498\) 0 0
\(499\) − 1.27745e12i − 0.922344i −0.887311 0.461172i \(-0.847429\pi\)
0.887311 0.461172i \(-0.152571\pi\)
\(500\) 0 0
\(501\) − 1.45745e12i − 1.03353i
\(502\) 0 0
\(503\) −2.40613e11 −0.167596 −0.0837979 0.996483i \(-0.526705\pi\)
−0.0837979 + 0.996483i \(0.526705\pi\)
\(504\) 0 0
\(505\) −3.01143e12 −2.06045
\(506\) 0 0
\(507\) − 7.94168e11i − 0.533798i
\(508\) 0 0
\(509\) − 5.46994e11i − 0.361204i −0.983556 0.180602i \(-0.942195\pi\)
0.983556 0.180602i \(-0.0578046\pi\)
\(510\) 0 0
\(511\) 2.90651e12 1.88572
\(512\) 0 0
\(513\) −1.94957e11 −0.124283
\(514\) 0 0
\(515\) − 1.13753e12i − 0.712574i
\(516\) 0 0
\(517\) − 2.21131e12i − 1.36127i
\(518\) 0 0
\(519\) 1.34426e12 0.813259
\(520\) 0 0
\(521\) 2.49243e12 1.48202 0.741009 0.671495i \(-0.234347\pi\)
0.741009 + 0.671495i \(0.234347\pi\)
\(522\) 0 0
\(523\) − 1.54454e12i − 0.902697i −0.892348 0.451349i \(-0.850943\pi\)
0.892348 0.451349i \(-0.149057\pi\)
\(524\) 0 0
\(525\) − 1.73167e12i − 0.994829i
\(526\) 0 0
\(527\) 5.02202e11 0.283616
\(528\) 0 0
\(529\) −1.12753e12 −0.626004
\(530\) 0 0
\(531\) 6.14299e11i 0.335316i
\(532\) 0 0
\(533\) 8.92846e11i 0.479186i
\(534\) 0 0
\(535\) −5.58079e12 −2.94513
\(536\) 0 0
\(537\) 1.42854e12 0.741322
\(538\) 0 0
\(539\) − 1.12498e12i − 0.574110i
\(540\) 0 0
\(541\) − 4.66468e11i − 0.234118i −0.993125 0.117059i \(-0.962653\pi\)
0.993125 0.117059i \(-0.0373466\pi\)
\(542\) 0 0
\(543\) 3.87904e10 0.0191481
\(544\) 0 0
\(545\) 1.02830e12 0.499269
\(546\) 0 0
\(547\) 1.63835e12i 0.782462i 0.920293 + 0.391231i \(0.127951\pi\)
−0.920293 + 0.391231i \(0.872049\pi\)
\(548\) 0 0
\(549\) − 2.53511e11i − 0.119103i
\(550\) 0 0
\(551\) −1.62916e12 −0.752978
\(552\) 0 0
\(553\) −3.99195e12 −1.81519
\(554\) 0 0
\(555\) − 3.78460e12i − 1.69318i
\(556\) 0 0
\(557\) 6.77404e11i 0.298194i 0.988823 + 0.149097i \(0.0476367\pi\)
−0.988823 + 0.149097i \(0.952363\pi\)
\(558\) 0 0
\(559\) −2.66647e11 −0.115501
\(560\) 0 0
\(561\) −9.62194e11 −0.410138
\(562\) 0 0
\(563\) − 2.75360e12i − 1.15508i −0.816361 0.577542i \(-0.804012\pi\)
0.816361 0.577542i \(-0.195988\pi\)
\(564\) 0 0
\(565\) 2.05785e12i 0.849563i
\(566\) 0 0
\(567\) −3.29659e11 −0.133949
\(568\) 0 0
\(569\) 2.14461e12 0.857716 0.428858 0.903372i \(-0.358916\pi\)
0.428858 + 0.903372i \(0.358916\pi\)
\(570\) 0 0
\(571\) − 3.43875e12i − 1.35375i −0.736098 0.676875i \(-0.763334\pi\)
0.736098 0.676875i \(-0.236666\pi\)
\(572\) 0 0
\(573\) − 1.64358e12i − 0.636934i
\(574\) 0 0
\(575\) 2.29121e12 0.874098
\(576\) 0 0
\(577\) −2.85300e12 −1.07155 −0.535773 0.844362i \(-0.679980\pi\)
−0.535773 + 0.844362i \(0.679980\pi\)
\(578\) 0 0
\(579\) − 5.09532e11i − 0.188416i
\(580\) 0 0
\(581\) 4.54154e12i 1.65353i
\(582\) 0 0
\(583\) −2.44256e12 −0.875663
\(584\) 0 0
\(585\) −4.04212e11 −0.142695
\(586\) 0 0
\(587\) − 2.36076e10i − 0.00820691i −0.999992 0.00410346i \(-0.998694\pi\)
0.999992 0.00410346i \(-0.00130617\pi\)
\(588\) 0 0
\(589\) 9.53732e11i 0.326518i
\(590\) 0 0
\(591\) 5.00523e11 0.168764
\(592\) 0 0
\(593\) −2.05586e12 −0.682727 −0.341364 0.939931i \(-0.610889\pi\)
−0.341364 + 0.939931i \(0.610889\pi\)
\(594\) 0 0
\(595\) − 3.22231e12i − 1.05400i
\(596\) 0 0
\(597\) 1.60604e12i 0.517453i
\(598\) 0 0
\(599\) −5.72088e12 −1.81569 −0.907846 0.419304i \(-0.862274\pi\)
−0.907846 + 0.419304i \(0.862274\pi\)
\(600\) 0 0
\(601\) 3.10411e12 0.970516 0.485258 0.874371i \(-0.338726\pi\)
0.485258 + 0.874371i \(0.338726\pi\)
\(602\) 0 0
\(603\) 1.18185e12i 0.364028i
\(604\) 0 0
\(605\) − 3.10121e12i − 0.941091i
\(606\) 0 0
\(607\) −1.37507e12 −0.411127 −0.205564 0.978644i \(-0.565903\pi\)
−0.205564 + 0.978644i \(0.565903\pi\)
\(608\) 0 0
\(609\) −2.75480e12 −0.811544
\(610\) 0 0
\(611\) − 1.01705e12i − 0.295227i
\(612\) 0 0
\(613\) 6.69362e12i 1.91465i 0.289016 + 0.957324i \(0.406672\pi\)
−0.289016 + 0.957324i \(0.593328\pi\)
\(614\) 0 0
\(615\) −5.56974e12 −1.56999
\(616\) 0 0
\(617\) 4.17726e11 0.116040 0.0580201 0.998315i \(-0.481521\pi\)
0.0580201 + 0.998315i \(0.481521\pi\)
\(618\) 0 0
\(619\) 6.34992e12i 1.73844i 0.494423 + 0.869221i \(0.335379\pi\)
−0.494423 + 0.869221i \(0.664621\pi\)
\(620\) 0 0
\(621\) − 4.36178e11i − 0.117693i
\(622\) 0 0
\(623\) −8.10445e11 −0.215540
\(624\) 0 0
\(625\) −1.47394e12 −0.386385
\(626\) 0 0
\(627\) − 1.82730e12i − 0.472179i
\(628\) 0 0
\(629\) − 4.14348e12i − 1.05545i
\(630\) 0 0
\(631\) −4.73127e12 −1.18808 −0.594040 0.804435i \(-0.702468\pi\)
−0.594040 + 0.804435i \(0.702468\pi\)
\(632\) 0 0
\(633\) 4.35598e12 1.07837
\(634\) 0 0
\(635\) 5.58263e12i 1.36256i
\(636\) 0 0
\(637\) − 5.17411e11i − 0.124511i
\(638\) 0 0
\(639\) 1.01163e12 0.240032
\(640\) 0 0
\(641\) 3.90673e12 0.914014 0.457007 0.889463i \(-0.348921\pi\)
0.457007 + 0.889463i \(0.348921\pi\)
\(642\) 0 0
\(643\) − 2.06347e12i − 0.476046i −0.971260 0.238023i \(-0.923501\pi\)
0.971260 0.238023i \(-0.0764993\pi\)
\(644\) 0 0
\(645\) − 1.66340e12i − 0.378423i
\(646\) 0 0
\(647\) −4.14405e12 −0.929728 −0.464864 0.885382i \(-0.653897\pi\)
−0.464864 + 0.885382i \(0.653897\pi\)
\(648\) 0 0
\(649\) −5.75773e12 −1.27394
\(650\) 0 0
\(651\) 1.61269e12i 0.351915i
\(652\) 0 0
\(653\) 8.11599e12i 1.74676i 0.487044 + 0.873378i \(0.338075\pi\)
−0.487044 + 0.873378i \(0.661925\pi\)
\(654\) 0 0
\(655\) −3.67010e12 −0.779098
\(656\) 0 0
\(657\) 2.49011e12 0.521403
\(658\) 0 0
\(659\) 8.57180e11i 0.177047i 0.996074 + 0.0885234i \(0.0282148\pi\)
−0.996074 + 0.0885234i \(0.971785\pi\)
\(660\) 0 0
\(661\) − 1.57325e12i − 0.320546i −0.987073 0.160273i \(-0.948762\pi\)
0.987073 0.160273i \(-0.0512375\pi\)
\(662\) 0 0
\(663\) −4.42541e11 −0.0889494
\(664\) 0 0
\(665\) 6.11948e12 1.21344
\(666\) 0 0
\(667\) − 3.64493e12i − 0.713055i
\(668\) 0 0
\(669\) − 3.41370e12i − 0.658882i
\(670\) 0 0
\(671\) 2.37612e12 0.452498
\(672\) 0 0
\(673\) 4.91142e12 0.922867 0.461434 0.887175i \(-0.347335\pi\)
0.461434 + 0.887175i \(0.347335\pi\)
\(674\) 0 0
\(675\) − 1.48358e12i − 0.275071i
\(676\) 0 0
\(677\) − 4.67942e12i − 0.856137i −0.903746 0.428069i \(-0.859194\pi\)
0.903746 0.428069i \(-0.140806\pi\)
\(678\) 0 0
\(679\) −2.23045e12 −0.402697
\(680\) 0 0
\(681\) 5.03329e12 0.896787
\(682\) 0 0
\(683\) − 5.92535e12i − 1.04189i −0.853591 0.520944i \(-0.825580\pi\)
0.853591 0.520944i \(-0.174420\pi\)
\(684\) 0 0
\(685\) 1.24394e11i 0.0215870i
\(686\) 0 0
\(687\) 1.85481e12 0.317683
\(688\) 0 0
\(689\) −1.12341e12 −0.189911
\(690\) 0 0
\(691\) − 4.43776e12i − 0.740479i −0.928936 0.370239i \(-0.879276\pi\)
0.928936 0.370239i \(-0.120724\pi\)
\(692\) 0 0
\(693\) − 3.08984e12i − 0.508905i
\(694\) 0 0
\(695\) −1.27883e13 −2.07912
\(696\) 0 0
\(697\) −6.09789e12 −0.978661
\(698\) 0 0
\(699\) 5.21956e12i 0.826965i
\(700\) 0 0
\(701\) − 9.55720e12i − 1.49486i −0.664343 0.747428i \(-0.731288\pi\)
0.664343 0.747428i \(-0.268712\pi\)
\(702\) 0 0
\(703\) 7.86888e12 1.21511
\(704\) 0 0
\(705\) 6.34454e12 0.967273
\(706\) 0 0
\(707\) − 1.05874e13i − 1.59369i
\(708\) 0 0
\(709\) 5.67373e12i 0.843258i 0.906769 + 0.421629i \(0.138541\pi\)
−0.906769 + 0.421629i \(0.861459\pi\)
\(710\) 0 0
\(711\) −3.42004e12 −0.501901
\(712\) 0 0
\(713\) −2.13379e12 −0.309207
\(714\) 0 0
\(715\) − 3.78862e12i − 0.542130i
\(716\) 0 0
\(717\) 1.50049e12i 0.212030i
\(718\) 0 0
\(719\) 3.80229e11 0.0530598 0.0265299 0.999648i \(-0.491554\pi\)
0.0265299 + 0.999648i \(0.491554\pi\)
\(720\) 0 0
\(721\) 3.99927e12 0.551152
\(722\) 0 0
\(723\) 6.96161e12i 0.947518i
\(724\) 0 0
\(725\) − 1.23976e13i − 1.66654i
\(726\) 0 0
\(727\) 1.32242e13 1.75576 0.877880 0.478881i \(-0.158957\pi\)
0.877880 + 0.478881i \(0.158957\pi\)
\(728\) 0 0
\(729\) −2.82430e11 −0.0370370
\(730\) 0 0
\(731\) − 1.82113e12i − 0.235892i
\(732\) 0 0
\(733\) 1.53563e13i 1.96480i 0.186797 + 0.982399i \(0.440189\pi\)
−0.186797 + 0.982399i \(0.559811\pi\)
\(734\) 0 0
\(735\) 3.22771e12 0.407945
\(736\) 0 0
\(737\) −1.10773e13 −1.38303
\(738\) 0 0
\(739\) − 6.82404e12i − 0.841669i −0.907137 0.420835i \(-0.861737\pi\)
0.907137 0.420835i \(-0.138263\pi\)
\(740\) 0 0
\(741\) − 8.40430e11i − 0.102405i
\(742\) 0 0
\(743\) 6.28798e12 0.756940 0.378470 0.925614i \(-0.376450\pi\)
0.378470 + 0.925614i \(0.376450\pi\)
\(744\) 0 0
\(745\) −2.31610e13 −2.75458
\(746\) 0 0
\(747\) 3.89089e12i 0.457200i
\(748\) 0 0
\(749\) − 1.96207e13i − 2.27796i
\(750\) 0 0
\(751\) 1.19773e13 1.37397 0.686986 0.726671i \(-0.258933\pi\)
0.686986 + 0.726671i \(0.258933\pi\)
\(752\) 0 0
\(753\) −3.11700e12 −0.353313
\(754\) 0 0
\(755\) 8.82315e12i 0.988240i
\(756\) 0 0
\(757\) 5.00435e12i 0.553881i 0.960887 + 0.276940i \(0.0893205\pi\)
−0.960887 + 0.276940i \(0.910680\pi\)
\(758\) 0 0
\(759\) 4.08823e12 0.447144
\(760\) 0 0
\(761\) 2.44206e12 0.263952 0.131976 0.991253i \(-0.457868\pi\)
0.131976 + 0.991253i \(0.457868\pi\)
\(762\) 0 0
\(763\) 3.61524e12i 0.386168i
\(764\) 0 0
\(765\) − 2.76066e12i − 0.291431i
\(766\) 0 0
\(767\) −2.64815e12 −0.276289
\(768\) 0 0
\(769\) 8.52460e12 0.879034 0.439517 0.898234i \(-0.355150\pi\)
0.439517 + 0.898234i \(0.355150\pi\)
\(770\) 0 0
\(771\) − 9.99776e12i − 1.01896i
\(772\) 0 0
\(773\) − 5.84609e12i − 0.588922i −0.955664 0.294461i \(-0.904860\pi\)
0.955664 0.294461i \(-0.0951401\pi\)
\(774\) 0 0
\(775\) −7.25769e12 −0.722672
\(776\) 0 0
\(777\) 1.33057e13 1.30962
\(778\) 0 0
\(779\) − 1.15805e13i − 1.12670i
\(780\) 0 0
\(781\) 9.48190e12i 0.911939i
\(782\) 0 0
\(783\) −2.36013e12 −0.224392
\(784\) 0 0
\(785\) −1.28838e12 −0.121096
\(786\) 0 0
\(787\) − 1.32719e13i − 1.23324i −0.787262 0.616618i \(-0.788502\pi\)
0.787262 0.616618i \(-0.211498\pi\)
\(788\) 0 0
\(789\) 1.91814e12i 0.176211i
\(790\) 0 0
\(791\) −7.23489e12 −0.657109
\(792\) 0 0
\(793\) 1.09285e12 0.0981363
\(794\) 0 0
\(795\) − 7.00802e12i − 0.622219i
\(796\) 0 0
\(797\) − 2.02632e12i − 0.177887i −0.996037 0.0889436i \(-0.971651\pi\)
0.996037 0.0889436i \(-0.0283491\pi\)
\(798\) 0 0
\(799\) 6.94616e12 0.602954
\(800\) 0 0
\(801\) −6.94335e11 −0.0595968
\(802\) 0 0
\(803\) 2.33394e13i 1.98093i
\(804\) 0 0
\(805\) 1.36911e13i 1.14910i
\(806\) 0 0
\(807\) −4.32037e12 −0.358584
\(808\) 0 0
\(809\) −3.71860e12 −0.305219 −0.152609 0.988287i \(-0.548768\pi\)
−0.152609 + 0.988287i \(0.548768\pi\)
\(810\) 0 0
\(811\) 8.32783e12i 0.675987i 0.941149 + 0.337993i \(0.109748\pi\)
−0.941149 + 0.337993i \(0.890252\pi\)
\(812\) 0 0
\(813\) 8.58916e12i 0.689515i
\(814\) 0 0
\(815\) 3.72951e13 2.96103
\(816\) 0 0
\(817\) 3.45851e12 0.271575
\(818\) 0 0
\(819\) − 1.42111e12i − 0.110370i
\(820\) 0 0
\(821\) 3.72889e12i 0.286441i 0.989691 + 0.143220i \(0.0457458\pi\)
−0.989691 + 0.143220i \(0.954254\pi\)
\(822\) 0 0
\(823\) −4.95662e12 −0.376605 −0.188303 0.982111i \(-0.560299\pi\)
−0.188303 + 0.982111i \(0.560299\pi\)
\(824\) 0 0
\(825\) 1.39054e13 1.04506
\(826\) 0 0
\(827\) − 2.05532e12i − 0.152794i −0.997077 0.0763968i \(-0.975658\pi\)
0.997077 0.0763968i \(-0.0243416\pi\)
\(828\) 0 0
\(829\) − 4.49719e12i − 0.330709i −0.986234 0.165354i \(-0.947123\pi\)
0.986234 0.165354i \(-0.0528768\pi\)
\(830\) 0 0
\(831\) −1.13387e12 −0.0824818
\(832\) 0 0
\(833\) 3.53378e12 0.254294
\(834\) 0 0
\(835\) 3.91936e13i 2.79014i
\(836\) 0 0
\(837\) 1.38165e12i 0.0973045i
\(838\) 0 0
\(839\) 9.05888e12 0.631169 0.315585 0.948897i \(-0.397799\pi\)
0.315585 + 0.948897i \(0.397799\pi\)
\(840\) 0 0
\(841\) −5.21531e12 −0.359499
\(842\) 0 0
\(843\) 7.82924e12i 0.533944i
\(844\) 0 0
\(845\) 2.13567e13i 1.44105i
\(846\) 0 0
\(847\) 1.09031e13 0.727903
\(848\) 0 0
\(849\) 1.31465e13 0.868411
\(850\) 0 0
\(851\) 1.76051e13i 1.15068i
\(852\) 0 0
\(853\) 2.43668e13i 1.57589i 0.615743 + 0.787947i \(0.288856\pi\)
−0.615743 + 0.787947i \(0.711144\pi\)
\(854\) 0 0
\(855\) 5.24276e12 0.335516
\(856\) 0 0
\(857\) 1.54772e13 0.980117 0.490059 0.871689i \(-0.336975\pi\)
0.490059 + 0.871689i \(0.336975\pi\)
\(858\) 0 0
\(859\) 1.56032e12i 0.0977790i 0.998804 + 0.0488895i \(0.0155682\pi\)
−0.998804 + 0.0488895i \(0.984432\pi\)
\(860\) 0 0
\(861\) − 1.95818e13i − 1.21434i
\(862\) 0 0
\(863\) −1.09332e13 −0.670962 −0.335481 0.942047i \(-0.608899\pi\)
−0.335481 + 0.942047i \(0.608899\pi\)
\(864\) 0 0
\(865\) −3.61496e13 −2.19549
\(866\) 0 0
\(867\) 6.58318e12i 0.395685i
\(868\) 0 0
\(869\) − 3.20555e13i − 1.90684i
\(870\) 0 0
\(871\) −5.09478e12 −0.299947
\(872\) 0 0
\(873\) −1.91090e12 −0.111346
\(874\) 0 0
\(875\) 1.39872e13i 0.806668i
\(876\) 0 0
\(877\) 1.51832e13i 0.866693i 0.901227 + 0.433347i \(0.142667\pi\)
−0.901227 + 0.433347i \(0.857333\pi\)
\(878\) 0 0
\(879\) −9.91156e11 −0.0560005
\(880\) 0 0
\(881\) 2.87715e13 1.60905 0.804527 0.593916i \(-0.202419\pi\)
0.804527 + 0.593916i \(0.202419\pi\)
\(882\) 0 0
\(883\) 2.30013e13i 1.27329i 0.771155 + 0.636647i \(0.219679\pi\)
−0.771155 + 0.636647i \(0.780321\pi\)
\(884\) 0 0
\(885\) − 1.65197e13i − 0.905225i
\(886\) 0 0
\(887\) 1.27341e13 0.690735 0.345367 0.938468i \(-0.387754\pi\)
0.345367 + 0.938468i \(0.387754\pi\)
\(888\) 0 0
\(889\) −1.96271e13 −1.05390
\(890\) 0 0
\(891\) − 2.64717e12i − 0.140712i
\(892\) 0 0
\(893\) 1.31914e13i 0.694162i
\(894\) 0 0
\(895\) −3.84161e13 −2.00129
\(896\) 0 0
\(897\) 1.88030e12 0.0969752
\(898\) 0 0
\(899\) 1.15458e13i 0.589528i
\(900\) 0 0
\(901\) − 7.67255e12i − 0.387863i
\(902\) 0 0
\(903\) 5.84809e12 0.292698
\(904\) 0 0
\(905\) −1.04315e12 −0.0516924
\(906\) 0 0
\(907\) 2.44366e12i 0.119897i 0.998201 + 0.0599484i \(0.0190936\pi\)
−0.998201 + 0.0599484i \(0.980906\pi\)
\(908\) 0 0
\(909\) − 9.07061e12i − 0.440655i
\(910\) 0 0
\(911\) 3.69051e12 0.177523 0.0887613 0.996053i \(-0.471709\pi\)
0.0887613 + 0.996053i \(0.471709\pi\)
\(912\) 0 0
\(913\) −3.64688e13 −1.73701
\(914\) 0 0
\(915\) 6.81739e12i 0.321531i
\(916\) 0 0
\(917\) − 1.29032e13i − 0.602607i
\(918\) 0 0
\(919\) 1.04045e13 0.481173 0.240587 0.970628i \(-0.422660\pi\)
0.240587 + 0.970628i \(0.422660\pi\)
\(920\) 0 0
\(921\) −8.41025e12 −0.385160
\(922\) 0 0
\(923\) 4.36100e12i 0.197778i
\(924\) 0 0
\(925\) 5.98805e13i 2.68935i
\(926\) 0 0
\(927\) 3.42631e12 0.152394
\(928\) 0 0
\(929\) −3.78338e13 −1.66652 −0.833258 0.552885i \(-0.813527\pi\)
−0.833258 + 0.552885i \(0.813527\pi\)
\(930\) 0 0
\(931\) 6.71100e12i 0.292761i
\(932\) 0 0
\(933\) − 1.58531e12i − 0.0684930i
\(934\) 0 0
\(935\) 2.58752e13 1.10721
\(936\) 0 0
\(937\) −7.67680e12 −0.325351 −0.162675 0.986680i \(-0.552012\pi\)
−0.162675 + 0.986680i \(0.552012\pi\)
\(938\) 0 0
\(939\) − 1.91566e13i − 0.804123i
\(940\) 0 0
\(941\) − 1.98175e12i − 0.0823941i −0.999151 0.0411970i \(-0.986883\pi\)
0.999151 0.0411970i \(-0.0131171\pi\)
\(942\) 0 0
\(943\) 2.59091e13 1.06696
\(944\) 0 0
\(945\) 8.86514e12 0.361612
\(946\) 0 0
\(947\) 1.68460e13i 0.680647i 0.940308 + 0.340324i \(0.110537\pi\)
−0.940308 + 0.340324i \(0.889463\pi\)
\(948\) 0 0
\(949\) 1.07345e13i 0.429618i
\(950\) 0 0
\(951\) 1.25150e13 0.496157
\(952\) 0 0
\(953\) 2.24737e13 0.882585 0.441293 0.897363i \(-0.354520\pi\)
0.441293 + 0.897363i \(0.354520\pi\)
\(954\) 0 0
\(955\) 4.41989e13i 1.71948i
\(956\) 0 0
\(957\) − 2.21211e13i − 0.852518i
\(958\) 0 0
\(959\) −4.37338e11 −0.0166968
\(960\) 0 0
\(961\) −1.96806e13 −0.744359
\(962\) 0 0
\(963\) − 1.68097e13i − 0.629856i
\(964\) 0 0
\(965\) 1.37023e13i 0.508651i
\(966\) 0 0
\(967\) −3.69178e13 −1.35774 −0.678871 0.734258i \(-0.737530\pi\)
−0.678871 + 0.734258i \(0.737530\pi\)
\(968\) 0 0
\(969\) 5.73991e12 0.209145
\(970\) 0 0
\(971\) 1.73104e13i 0.624913i 0.949932 + 0.312456i \(0.101152\pi\)
−0.949932 + 0.312456i \(0.898848\pi\)
\(972\) 0 0
\(973\) − 4.49604e13i − 1.60813i
\(974\) 0 0
\(975\) 6.39549e12 0.226649
\(976\) 0 0
\(977\) 2.58899e13 0.909085 0.454543 0.890725i \(-0.349803\pi\)
0.454543 + 0.890725i \(0.349803\pi\)
\(978\) 0 0
\(979\) − 6.50790e12i − 0.226422i
\(980\) 0 0
\(981\) 3.09730e12i 0.106776i
\(982\) 0 0
\(983\) 2.22442e13 0.759848 0.379924 0.925018i \(-0.375950\pi\)
0.379924 + 0.925018i \(0.375950\pi\)
\(984\) 0 0
\(985\) −1.34600e13 −0.455599
\(986\) 0 0
\(987\) 2.23058e13i 0.748154i
\(988\) 0 0
\(989\) 7.73774e12i 0.257176i
\(990\) 0 0
\(991\) 3.85698e13 1.27033 0.635165 0.772377i \(-0.280932\pi\)
0.635165 + 0.772377i \(0.280932\pi\)
\(992\) 0 0
\(993\) 3.78465e12 0.123525
\(994\) 0 0
\(995\) − 4.31894e13i − 1.39693i
\(996\) 0 0
\(997\) − 1.29804e13i − 0.416062i −0.978122 0.208031i \(-0.933294\pi\)
0.978122 0.208031i \(-0.0667056\pi\)
\(998\) 0 0
\(999\) 1.13995e13 0.362109
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 384.10.d.a.193.4 8
4.3 odd 2 384.10.d.b.193.8 yes 8
8.3 odd 2 384.10.d.b.193.1 yes 8
8.5 even 2 inner 384.10.d.a.193.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.d.a.193.4 8 1.1 even 1 trivial
384.10.d.a.193.5 yes 8 8.5 even 2 inner
384.10.d.b.193.1 yes 8 8.3 odd 2
384.10.d.b.193.8 yes 8 4.3 odd 2